Answer 1

Core as in what you learned here or what Oracle defines as core. For the former, the short answer is that it is unlikely. The latter, yes you can.
The reason you can with the latter is because it includes various packages and frameworks. With knowledge of those, you can do a variety of practical things; web development, application making etc.
That is what the core java on this app lacks. It does not teach you about networking, how to make GUI applications, security, web services etc. Knowledge of a few of those would make you eligible for alot of jobs online and off. So you just need to add to the basic knowledge you have of java. Learn more things, become more advanced and become able to do more things.